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Triple-Girder Model for Modal Analysis of Cable-Stayed Bridges with Warping Effect L.D

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Triple-Girder Model for Modal Analysis of Cable-Stayed Bridges with Warping Effect L.D http://www.paper.edu.cn Engineering Structures 22 (2000) 1313–1323 www.elsevier.com/locate/engstruct Triple-girder model for modal analysis of cable-stayed bridges with warping effect L.D. Zhu a, H.F. Xiang a, Y.L. Xu b,* a College of Civil Engineering, Tongji University, Shanghai 200092, People’s Republic of China b Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received 9 February 1999; received in revised form 19 July 1999; accepted 29 July 1999 Abstract In the modal analysis of cable-stayed bridges using the finite element approach, the single-girder model or the double-girder model is often adopted for modeling the bridge deck. These two models are simple and easy to embody in commercial software packages, but the warping stiffness (constant) of bridge deck cannot be properly taken into consideration. This paper thus presents a triple-girder model consisting of one central girder and two side girders symmetrically connected to each other by transverse links. The proposed triple-girder model can easily consider the warping stiffness and other section properties of the bridge deck, and at the same time keep the user-friendly features in the single-girder model. The Nanpu cable-stayed bridge, built recently in China, is then taken as a case study to verify the rationality of the proposed triple-girder model through a comparison with measured data. The study shows that the modal properties of the cable-stayed bridge obtained from the modal analysis using the triple-girder model are more reasonable and close to the measured data. Further studies of four more cable-stayed bridges in China highlight that the extent of the warping effect depends greatly on the type of bridge deck and on the type of bridge tower–cable system. A simple approach for estimating some section properties of typical open-section decks is also suggested to further facilitate the use of the triple-girder model. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Cable-stayed bridge; Modal analysis; Triple-girder model; Warping constant; Case study 1. Introduction walled element model have been developed to model the bridge deck [1–7]. Increasing attention has been paid to the dynamic In the single-girder beam element model, the bridge design of cable-stayed bridges in recent years as this type deck is represented by a single beam and the cross-sec- of bridge becomes more and more popular. The dynamic tion properties of the bridge deck are assigned to the design of cable-stayed bridges subject to wind and earth- beam as equivalent properties. Conventional beam quake loading depends largely on knowledge of the elements of 12 degrees of freedom are normally used. bridges’ modal properties. Modal analysis is therefore This model is suitable for cable-stayed bridges of rela- the first important step towards a successful dynamic tively large pure torsional stiffness so that warping stiff- design. Since modern cable-stayed bridges involve a var- ness can be neglected in the equivalent beam. For cable- iety of decks, towers and cables that are connected stayed bridges with double cable planes and an open- together in different ways, the finite element method is section deck, the pure torsional stiffness of the bridge commonly regarded as the most proper way for con- deck may be small and the warping stiffness may ducting the modal analysis. In this connection, the sin- become important for the modal properties of the bridge. gle-girder beam element model, the double-girder beam This, however, presents a difficult task for the single- element model, the shell element model and the thin- girder model. To take the warping stiffness into account in the single-girder beam element model, Wilson and Gravelle [8] introduced an equivalent pure torsional con- eq * Corresponding author. Tel.: + 852-2766-6050; fax: + 852- stant Jd such that: 2334-6389. ϭ eq⌽Ј ϭ ⌽Ј Ϫ ⌽ЈЈЈ E-mail address: [email protected] (Y.L. Xu). T GJd GJd EJw , (1) 0141-0296/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S0141-0296(99)00077-2 转载 中国科技论文在线 http://www.paper.edu.cn 1314 L.D. Zhu et al. / Engineering Structures 22 (2000) 1313–1323 where T is the applied torsional moment, Jd is the pure grams to carry out the modal analysis of cable-stayed torsional constant of the transformed deck cross-section bridges that are not available in most commercial com- of one material, Jw is the warping constant of the trans- puter packages. Nevertheless, when analyzing wind- formed deck cross-section, E is the modulus of elasticity induced lateral–torsional buckling and flutter problems of the transformed material, G is the shear modulus of of long-span cable-stayed bridges, large twist defor- the transformed material, and ⌽ is the torsional mode mation of the bridge deck and thus geometric non-linear shape of the bridge deck. A prime indicates the first- analysis should be considered, so that thin-walled order derivative with respect to position coordinate. elements may have to be applied [7]. Shell elements are By assuming that torsional mode functions are sine sometimes used to model a bridge deck. In this case, functions, they derived the equivalent pure torsional con- warping effects can be properly considered [3]. How- stant as follows: ever, extremely large computational effort is required for long-span cable-stayed bridges. EJ np 2 eq ϭ ϩ wͩ ͪ To overcome the shortcomings in the various models Jd Jd , (2) G L mentioned above, this paper presents a triple-girder where n is the torsional mode number (n = 1, 2, %) and model consisting of one central girder and two side gir- L is the length of the main span of the bridge. Appar- ders connected by transverse links to include the warping ently, the equivalent pure torsional constant depends on stiffness. Conventional beam elements of 12 degrees of the torsional mode number. Wilson and Gravelle [8] cal- freedom are used to model all girders, and thus the user- culated the equivalent pure torsional constant for each friendly features in the single-girder beam element of the first three torsional modes and used the average model remain. The Nanpu cable-stayed bridge built as the final equivalent constant for the bridge deck. This recently in China is taken as a case study to verify the approximation, together with the assumption of a sine rationality of the proposed triple-girder model through a torsional mode function, result in inconvenience to the comparison with measured data. After the comparison, modal analysis and uncertainties in the modal proper- four more cable-stayed bridges are analyzed to investi- ties obtained. gate the relationships between the warping effect and In the double-girder beam element model, two equiv- the type of bridge structural system. A simple way of alent beams connected by transverse links are respect- calculating some section properties for typical open-sec- ively located in each cable plane [2]. The warping stiff- tion decks is then suggested to further facilitate use of ness of the bridge deck can be considered through the the triple-girder model. opposite vertical bending stiffness of the two girders. However, the vertical bending stiffness of the bridge deck should be taken also by the vertical bending stiff- 2. Triple-girder model ness of the two girders if the transverse links are rigid and do not provide any section properties. This treatment Let us consider a typical open cross-section of bridge then leads to uncertainties in the equivalence of warping deck in cable-stayed bridges with double cable planes stiffness and vertical bending stiffness, as well as in the [Fig. 1(a)]. The cross-section should be seen as the trans- equivalence of lateral bending stiffness and longitudinal formed section for one material obtained by well-known stiffness. If the transverse links are modeled as elastic techniques as discussed, for example, by Craig [9]. The members taking some section properties of the bridge section properties of the deck are defined as follows: A deck, the equivalence of the vertical, lateral and torsional is the cross-sectional area; A¯ y is the shear area in the y- stiffnesses will be difficult to execute, and the compu- axis; A¯ z is the shear area in the z-axis; Jy is the second tation efforts for modal analysis will be increased. Fur- moment of area about the neutral y-axis; Jz is the second thermore, because the double-girder beam element moment of area about the neutral z-axis; Jd is the pure model becomes a shear type of structure in the lateral torsional constant of the cross-section; and Jw is the direction, the properties and behavior of the original warping constant with respect to the shear center. The bridge deck will be distorted to some extent. material properties are denoted as: E is the modulus of Another approach to consider warping effects in elasticity of the transformed material; G is the shear modal analysis of cable-stayed bridges is to use thin- modulus of the transformed material. walled elements to model the bridge deck. Compared The aforementioned bridge deck is now represented with the conventional beam element, the thin-walled by a triple-girder model as shown in Figs. 1(b) and (c). element has an additional warping degree of freedom at In the triple-girder model, a central girder (No. 1) is each end, expressed by the rate of twist. The elastic and located at the centroid of the original bridge deck. Two geometric stiffness matrices with or without joint side girders (No.
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